2.1.1 Overview
Perhaps the first numerical approach to study the cosmic
censorship conjecture consisted of attempts to create naked
singularities. Many of these studies were motivated by Thorne's
``hoop conjecture'' [175] that collapse will yield a black hole only if a mass
M
is compressed to a region with circumference
in all directions. (As is discussed by Wald [178], one runs into difficulties in any attempt to formulate the
conjecture precisely. For example, how does one define
C
and
M, especially if the initial data are not at least axially
symmetric?) If the hoop conjecture is true, naked singularities
may form if collapse can yield
in some direction. The existence of a naked singularity is
inferred from the absence of an apparent horizon (AH) which can
be identified locally by following null geodesics. Although a
definitive identification of a naked singularity requires the
event horizon (EH) to be proven to be absent, to identify an EH
requires knowledge of the entire spacetime. While one finds an AH
within an EH [120,
121], it is possible to construct a spacetime slicing which avoids
the AH even though an EH is present [180]. Methods to find an EH in a numerically determined spacetime
have only recently become available and have not been applied to
this issue [132,
136].
In the best known attempt to produce naked singularities, Shapiro
and Teukolsky (ST) [169] considered collapse of prolate spheroids of collisionless gas.
(Nakamura and Sato [146] had previously studied the collapse of nonrotating deformed
stars with an initial large reduction of internal energy and
apparently found spindle or pancake singularities in extreme
cases.) ST solved the general relativistic Vlasov equation for
the particles along with Einstein's equations for the
gravitational field. Null geodesics were followed to identify an
AH if present. The curvature invariant
was also computed. They found that an AH (and presumably a BH)
formed if
everywhere but no AH (and presumably a naked singularity) in the
opposite case. In the latter case, the evolution (not
surprisingly) could not proceed past the moment of formation of
the singularity. In a subsequent study, ST [170] also showed that a small amount of rotation (counter rotating
particles with no net angular momentum) does not prevent the
formation of a naked spindle singularity. However, Wald and Iyer
[180] have shown that the Schwarzschild solution has a time slicing
whose evolution approaches arbitrarily close to the singularity
with no AH in any slice, (but of course, with an EH in the
spacetime). This may mean that there is a chance that the
increasing prolateness found by ST in effect changes the slicing
to one with no apparent horizon just at the point required by the
hoop conjecture. While, on the face of it, this seems unlikely,
Tod gives an example where the AH does not form on a chosen
constant time slicebut rather different portions form at
different times. He argues that a numerical simulation might be
forced by the singularity to end before the formation of the AH
is complete. Such an AH would not be found by the simulations [177]. In response to such a possibility, Shapiro and Teukolsky
considered equilibrium sequences of prolate relativistic star
clusters [171]. The idea is to counter the possibility that an EH might form
after the simulation must stop. If an equilibrium configuration
is nonsingular, it cannot contain an EH, since singularity
theorems say that an EH implies a singularity. However, a
sequence of nonsingular equilibria with rising
I
ever closer to the spindle singularity would lend support to the
existence of a naked spindle singularity since one can approach
the singular state without formation of an EH. They constructed
this sequence and found that the singular end points were very
similar to their dynamical spindle singularity. Wald believes,
however, that it is likely that the ST slicing is such that their
singularities are not nakedthe AH is present but has not yet
appeared in their time slices [178].
Another numerical study of the hoop conjecture was made by
Chiba et al [58]. Rather than a dynamical collapse model, they searched for AH's
in analytic initial data for discs, annuli, and rings. Previous
studies of this type were done by Nakamura et al [147] with oblate and prolate spheroids and by Wojtkiewicz [182] with axisymmetric singular lines and rings. The summary of
their results is that an AH forms if
. (Analytic results due to Barrabès et al [5,
4] and Tod [177] give similar quantitative results with different initial data
classes and (possibly) definition of
C
.)
There is strong analytical evidence against the development of
spindle singularities. It has been shown by Chrusciel and
Moncrief that strong cosmic censorship holds in AF electrovac
solutions which admit a
symmetric Cauchy surface [24]. The evolutions of these highly nonlinear equations are in fact
nonsingular.
Motivated by ST's results [169], Echeverria [70] numerically studied the properties of the naked singularity
that is known to form in the collapse of an infinite, cylindrical
dust shell [175]. While the asymptotic state can be found analytically, the
approach to it must be followed numerically. The analytic
asymptotic solution can be matched to the numerical one (which
cannot be followed all the way to the collapse) to show that the
singularity is strong (an observer experiences infinite
stretching parallel to the symmetry axis and squeezing
perpendicular to the symmetry axis). A burst of gravitational
radiation emitted just prior to the formation of the singularity
stretches and squeezes in opposite directions to the singularity.
This result for dust conflicts with rigorously nonsingular
solutions for the electrovac case [24]. One wonders then if dust collapse gives any information about
singularities of the gravitational field.
Nakamura et al (NSN) [148] conjectured that even if naked spindle singularities could
exist, they would either disappear or become black holes. This
demise of the naked singularity would be caused by the back
reaction of the gravitational waves emitted by it. While NSN
proposed a numerical test of their conjecture, they believed it
to be beyond the current generation of computer technology.
The results of all these searches for naked spindle
singularities are controversial, but they could be resolved if
the presence or absence of the EH could be determined. One could
demonstrate numerically whether or not Wald's view of ST's
results is correct by using existing EH finders [132,
136] in a relevant simulation.

Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr19987
© MaxPlanckGesellschaft. ISSN 14338351
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